292 ANALYTICAL MECHANICS
point of the field of force. This fact enables us to state the conditions which determine the type of the orbit in the following forms:
I. When the velocity of projection equals the velocity
from infinity the orbit is a parabola, II. When the velocity of projection is less than the
velocity from infinity the orbit.is an ellipse. III. When the velocity of projection is greater than the velocity from infinity the orbit is a hyperbola.
Thus if a comet starts from rest at an infinite distance from the sun and falls towards the sun its orbit will be a parabola, If it is projected towards the sun from an infinite distance its orbit will be a hyperbola. If it falls from rest, starting from a finite distance, its orbit will be an ellipse.
223. Period of Revolution. — From equation (III) we have
= rco = r~cftT h dt = r • r dd = 2 dA,
where dA is the area swept over by the radius vector in th< time dt. Therefore when the orbit is an ellipse the period o:
T I dA (irab = area of ellipse^
where a and 6 are the semi-major axis and semi-minor axi of the ellipse, respectively. But by equations (6)
h = ^ep • fji ___
and by the properties of the ellipse ep = —, therefore h=y — ^